Transition State Theory

Example:  Calculating the Frequency Factor Using Transition State Theory

Use transition state theory to calculate the frequency factor A at 300K for the reaction

                                                              H + HBr ® H₂ + Br

Additional Information Literature values.† (Note:  Most of this information can be obtained from computational chemistry software packages such as Cerius², Spartan or Cache.)

Reactants.  H, HBr

            H atom (mass)                                   1 amu

            HBr (mass)                                        80.9 amu

            HBr vibration wave number               2650 cm^–1

            H – Br separation distance = 142 pm

Transition State Complex.  H–H–Br

            Vibration wave numbers

                     2340 cm^–1

                     460 cm^–1 (degenerate

            Separation distances

Solution

            The reaction is

                                                    H + HBr  H–H–Br ® H₂ + Br

            The specific reaction rate is

Reactants

            Hydrogen

            Rotation

            Total partition function

            Translation

            Vibration

            1)   u = 2340 cm^–1

            2)   u = 460 cm²

            Rotation

            Calculate the rotational partition function, , for the transition state shown below.

The rotational partition function is

The total partition function for the transition state is

We now calculate the frequency factoring A.

Data From Computational Chemistry

Now let’s calculate A and E using the parameters from cache.

            For the reaction:

                                                              H + HBr ® H₂ + Br

The 3D potential energy surfaces of the reacting particles along the reaction coordinates was calculated using the MOPAC PM3 method:

The transition state structure was found at the saddle point, refined by using the DFT/B88-PW91 method as:

In the transition state, the three atoms are linear and the H-Br distance is 1.48 Å while the H-H distance is 1.55 Å.

            The transition state was further proved by vibrational analysis (PM3 FORCE) showing one and only one negative vibration (imaginary frequency of crossing the barrier). Moreover, the negative vibration corresponds to the movement of the atoms on the two reaction coordinates.

Summary of Information from Cache Software

Reactants.  H, HBr

            H atom (mass)                                   1 amu

            HBr (mass)                                        80.9 amu

            HBr vibration wave number               2122 cm^–1

            H – Br separation distance = 147 pm

Transition State Complex.  H–H–Br

            Vibration wave numbers

                     1736 cm^–1

                     289 cm^–1

            Separation distances

                                                                      q_r = 137.5

Therefore, the standard enthalpy of activation is:

                                    kcal/mol = 9.2 kJ/mol

The intrinsic Arrhenius activation energy is:

                                     kcal/mol = 11.7 kJ/mol

Barrier height E₀ (difference between zero-point levels of activated complexes and reactants) (because the conversion between the a.u. and the kcal/mol units is very large, we need to maintain a high number of decimal point):

 kcal/mol = 2.7 kJ/mol

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